n sip 01 matrix gain talk

8/12/2019 n Sip 01 Matrix Gain Talk

http://slidepdf.com/reader/full/n-sip-01-matrix-gain-talk 1/13



2

OutlineOutline

• Digital halftoning– Modeling grayscale error diffusion halftoning– Modeling color error diffusion halftoning

• Matrix gain model for color error diffusion– Validating the signal shaping predicted by the model– Validating the noise shaping predicted by the model

• Conclusions



3

Grayscale Error DiffusionGrayscale Error Diffusion• Shape quantization noise into high frequencies

• Two-dimensional sigma-delta modulation

• Design of error filter is key to high quality

current pixel

weights

3/16

7/16

5/16 1/16

+ _

_+

e(m )

b (m ) x(m )

difference threshold

computeerror

shape error

u (m )

)(mh

Error Diffusion

Spectrum



4

Modeling Grayscale Error DiffusionModeling Grayscale Error Diffusion

• Sharpening is caused by a correlated error image [Knox, 1992]

Floyd-Steinberg

Jarvis

Error images Halftones



5

Modeling Grayscale Error DiffusionModeling Grayscale Error Diffusion

• Apply sigma-delta modulation analysis to two dimensions– Linear gain model for quantizer in 1-D [Ardalan and Paulos, 1988]

– Linear gain model for grayscale image [Kite, Evans, Bovik, 2000]

– Signal transfer function (STF) and noise transfer function (NTF)– 1 – H( z) is highpass so H( z) is lowpass

K s

u s(m )

Signal Path

K s u s(m )

K n

u n (m )

+

n (m )

K n u n (m ) + n (m)

Noise Path, K n =1

( ))(1

)(z

zz

H N

B NTF n

−==

( )( ) ( )zz

z H K

K

X

BSTF s

ss

11)( −+==

Q(.)u (m ) b (m ) {



6

Vector Color Error DiffusionVector Color Error Diffusion

• Error filter has matrix-valued coefficients

• Algorithm for adapting matrix coefficients[Akarun, Yardimci, Cetin 1997]

( ) ( )ÿ

( )vectormatrix

kmekhmtk

−= ÿ℘∈

+ _

_+

e(m )

b (m )x(m )

difference threshold

computeerror

shape error

u (m )

)(mh

t(m )



7

The Matrix Gain ModelThe Matrix Gain Model

• Replace scalar gain with a matrix

– Noise is uncorrelated with signal component of quantizer input– Convolution becomes matrix–vector multiplication in frequencydomain

( ) ( ) 12

minarg −

=−=uubuA

CCmuAmbK E s

IK =n

( ) ( ) ( )zNzHIzB −=n

( ) ( )( )( ) ( )zXIKzHIKzB 1−

−+=s

Noise component of output

Signal component of output

u (m ) quantizer inputb (m ) quantizer output



8

How to Construct an Undistorted HalftoneHow to Construct an Undistorted Halftone

• Pre-filter with inverse of signal transfer function to obtain

undistorted halftone

• Pre-filtering is equivalent to the following when

( ) ( )( )[ ] 1−−+= KIKzHIzG

IKL −= −1

Modified error diffusion

+ _

_+

e(m )

b (m )x(m )u (m )

)(mh

t(m )

L



9

Validation #1 by Constructing Undistorted HalftoneValidation #1 by Constructing Undistorted Halftone

• Generate linearly undistorted halftone

• Subtract original image from halftone

• Since halftone should be “undistorted”, the residual should

be uncorrelated with the original

=

0027.00011.00058.0

0020.00023.00054.0

0040.00009.00052.0

rxC

Correlation matrix of residualimage (undistorted halftone minusinput image) with the input image



10

Validation #2 by Knox’s ConjectureValidation #2 by Knox’s Conjecture

=

1836.02952.02063.0

1605.03295.02787.0

0999.02989.03204.0

exC

Correlation matrix for anerror image and input imagefor an error diffused halftone

=

0150.00142.00428.0

0164.00144.00493.0

0122.00235.00455.0

exC

Correlation matrix for anerror image and input imagefor an undistorted halftone

( ) 0=zE s

( ) ( )zNzE =n



11

Validation #3 by Distorting Original ImageValidation #3 by Distorting Original Image• Validation by constructing a

linearly distorted original

– Pass original image through

error diffusion with matrix

gain substituted for quantizer

– Subtract resulting color

image from color halftone

– Residual should be shaped

uncorrelated noise

Correlation matrix of residual

image (halftone minus distortedinput image) with the input image

=

0062.00040.00082.0

0049.00039.00065.0

0051.00007.00067.0

rxC



12

Validation #4 by Noise ShapingValidation #4 by Noise Shaping

• Noise process is error image for an undistorted halftone

• Use model noise transfer function to compute noise spectrum

• Subtract actual halftone from modeled halftone and compute

actual noise spectrum

0 0.5 1 1.510

15

20

25

30

35

40

radial frequency

2 0 * l o

g 1 0 (

e r r o r

)

red error spectra

observedpredicted

0 0.5 1 1.515

20

25

30

35

40

radial frequency

2 0

* l o

g 1 0 (

e r r o r

)

green error spectra

observedpredicted

0 0.5 1 1.515

20

25

30

35

radial frequency

2 0 * l o

g 1 0 (

e r r o r

)

blue error spectra

observedpredicted



13

ConclusionsConclusions

• Modeling of color error diffusion in the frequency domainusing a coupled matrix gain and noise injection approach

• Linearizes error diffusion

• Predicts linear distortion and noise shaping effects

• Signal frequency distortion may be ‘‘cancelled’’

• Filters may be designed for optimum noise shaping

n sip 01 matrix gain talk

Documents